Question: Perform the row operation, $-R_2\rightarrow R_2$, on the following matrix. $\left[\begin{array} {ccc} 4 & 1 & 1 & 7 \\ 7 & 0 & 4 & 0 \\ 2 & 3 & 9 & 0 \end{array} \right] $
Explanation: Background There are three basic row operations that can be performed on matrices. $R_i \leftrightarrow R_j$. This symbol tells us to interchange rows $i$ and $j$. $cR_i \rightarrow R_i$. This symbol tells us to multiply a row $i$ by a constant $c$. $R_i + cR_j \rightarrow R_i$. This symbol tells us to add $c$ times row $j$ to row $i$. Finding the new row to be used For the given matrix, $R_2$ is given below. $R_2=\left[\begin{array} {ccc} 7 & 0 & 4 & 0 \end{array} \right]$ We are asked to perform the row operation, $-R_2\rightarrow R_2$. Therefore, we must multiply $R_2$ by $-1$. $\begin{aligned}-R_2 &= -1\left[\begin{array} {ccc} 7 & 0 & 4 & 0 \end{array} \right] \\\\&=\left[\begin{array} {ccc} -7 & 0 & -4 & 0 \end{array} \right]\end{aligned}$ Substituting the row Now, we must substitute row $R_2$ with $-R_2$. $\left[\begin{array} {ccc} 4 & 1 & 1 & 7 \\ {7} & {0} & {4} & {0} \\ 2 & 3 & 9 & 0 \end{array} \right] \xrightarrow{-R_2\rightarrow R_2} \left[\begin{array} {ccc} 4 & 1 & 1 & 7 \\ {-7} & {0} & {-4} & {0} \\ 2 & 3 & 9 & 0 \end{array} \right] $ Summary Our resultant matrix is the following. $\left[\begin{array} {ccc} 4 & 1 & 1 & 7 \\ -7 & 0 & -4 & 0 \\ 2 & 3 & 9 & 0 \end{array} \right]$